Science - USA (2021-12-10)

fields, the antisymmetric peak detected in the
GMR resistance is directly proportional to the
derivative of the McClure peak with respect to
the chemical potential (controlled by the gate
voltage), as detailed in ( 14 ). The experimental
detection of this peak and its evolution with
magnetic field are the central result of our work.
Both the peak width and amplitude increase
linearly with field, as shown in Fig. 2, E and F.
Above 0.6 T,@M/@Vg(Vg) displays periodic
oscillations in addition to the antisymmetric
peak around the Dirac point. These oscilla-
tions are related to the expected de Haas–van
Alphen oscillations of the magnetization, as
discussed below.
The magnetization, shown in Fig. 2D, is
obtained by the integration of the curves in
Fig. 2A. The peak amplitude translates into
a few nanoteslas induced in the GMR plane
by graphene’s orbital response to a 0.1-T per-
pendicular field. This illustrates the sensitiv-
ity of our experiment. The correspondence
between this detected field,BGMR, and mag-
netization is obtained by modeling the orbital
magnetic moment as an effective current loop
whose geometry is defined by the gated re-
gion of graphene (fig. S10). We find that posi-
tive magnetic fields produce a negative peak
in magnetization, and vice versa, which is con-
sistent with the expected diamagnetic response
of graphene ( 2 ). The sign of the response was
carefully determined via the sign of the re-
sponse of the GMR sensor to a horizontal field
of known orientation. We can assert that the
signal cannot be attributed to gate voltage–
dependent magnetism of paramagnetic im-
purities, given the absence of temperature de-
pendence between 4.2 and 40 K ( 15 )(seefig.
S9). In addition, thanks to our gate modulation
technique, we can exclude spurious contribu-
tions from impurities or defects in alumina or
graphene, which would not depend on gate
voltage. This contrasts with all previous mea-
surements of graphene’s magnetism, which
were performed on large ensembles of flakes.
In the following, we compare our results to
theoretical predictions, taking into account the
variations of the chemical potential caused by
charge inhomogeneity, and ignoring the smaller
broadening due to temperature ( 14 ). Assuming
a Gaussian distribution for the electrochem-
ical potentialm′of standard deviations,

Psð Þ¼m′

1

ffiffiffiffiffi

2 p

p

s

exp 

m′^2

2 s^2



ð 2 Þ

yields a smoothed susceptibility,

csð Þ¼m ∫PsðÞm′c 0 ðÞmm′dm′ ð 3 Þ

Then, thed-peak of the susceptibility is broad-
ened as

csð Þ¼m

2 e^2 v^2 F

3 p

PsðÞm ð 4 Þ

SCIENCEscience.org 10 DECEMBER 2021•VOL 374 ISSUE 6573 1401

Fig. 3. Calculated chemical potential dependence of the orbital magnetization of graphene in a finite

magnetic field.The calculations are based on Eq. 6; see ( 14 ) for more details. (A) Evolution of the graphene

spectrum in a magnetic field [adapted from ( 3 )]. The condensation of the continuous spectrum into

Landau levels decreases the energy, except for the zero energy level whose contribution is predominant.

Globally, the net result is an increase of the energy with the field—that is, a diamagnetic response [see also

figure 5 of ( 3 )]. (B) Without disorder, the magnetization, plotted as a function of the rescaled chemical

potentialm/eB, exhibits discontinuities at the Landau level energies

ffiffiffi

n

p

eB; a.u., arbitrary units. (C) Sketch

illustrating the spatial distribution of electrochemical potentialsm′=mD–hmDiwheremDis the local

Dirac point andhmDiis its spatial average. (D) Rounding ofM(m/eB) by a Gaussian chemical potential

distribution with a variances= 0.1eB.(E) CalculatedM(m) for different magnetic fields fors= 50 K. At low

fields, the oscillations disappear and the magnetization displays a Gaussian diamagnetic peak atm= 0. This

peak is broadened by the magnetic field as soon aseB≥s.

AB

Fig. 4. Comparison of theory to experiment.Fit of detected AC magnetization response to a gate voltage

modulation of 50 mV, as a function of the DC gate voltage divided byae^2 B¼ 2 aeħv^2 FB. Dashed lines

show the theoretical gate dependence of@M/@Vg, withs 0 =165Kands∞= 50 K, including the extra

rounding effect owing to the 50-mV AC gate modulation. In (A), the amplitude of the theoretical signal has

been rescaled by a factor of 1/2.6 at 0.1 T and by a factor of 1/2 at 0.2 T to fit quantitatively the

experimental data. In (B), the rescaling factors are closer to unity for higher fields for which the McClure

peak is expected to be independent ofs 0.

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